3.2555 \(\int \frac{1}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 \sqrt{5 x+3}}{11 \sqrt{1-2 x}} \]

[Out]

(2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0019502, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ \frac{2 \sqrt{5 x+3}}{11 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]

[Out]

(2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{2 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}\\ \end{align*}

Mathematica [A]  time = 0.0033949, size = 22, normalized size = 1. \[ \frac{2 \sqrt{5 x+3}}{11 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]

[Out]

(2*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x])

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Maple [A]  time = 0.002, size = 17, normalized size = 0.8 \begin{align*}{\frac{2}{11}\sqrt{3+5\,x}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x)

[Out]

2/11*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 3.83902, size = 28, normalized size = 1.27 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{11 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

-2/11*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.68095, size = 62, normalized size = 2.82 \begin{align*} -\frac{2 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{11 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-2/11*sqrt(5*x + 3)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 1.04421, size = 48, normalized size = 2.18 \begin{align*} \begin{cases} \frac{\sqrt{10}}{11 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{\sqrt{10} i}{11 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Piecewise((sqrt(10)/(11*sqrt(-1 + 11/(10*(x + 3/5)))), 11/(10*Abs(x + 3/5)) > 1), (-sqrt(10)*I/(11*sqrt(1 - 11
/(10*(x + 3/5)))), True))

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Giac [A]  time = 2.60861, size = 35, normalized size = 1.59 \begin{align*} -\frac{2 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{55 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-2/55*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)